curve #4722

y² = x³ − 3778726822874847152100x

a-invariants: [0, 0, 0, -3778726822874847152100, 0]
c-invariants: [181378887497992663300800, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 60459629165997554433600
discriminant (Δ): 3453158104815629626844924644220676235650336693235057571440704000000
Faltings height: 11.4570
naive height: 160.6646
primes of bad reduction: 2, 3, 5, 11, 19, 41, 43, 67, 83
regulator: 334590.55300819752770042581248724913386342823679286
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:01 UTC
last updated: 2026-10-20 21:26:01 UTC

Witness: 6 independent points log in to add more points →

(357359562285, 210443685836202225)
(-46277294910, 8704157339770200)
(4694189254194/25, 9610000571457007128/125)
(-7597027350, 5316825173061600)
(4364802232681/16, 8884464763369266379/64)
(6230972994444/25, 15073220074408593372/125)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 61471349610. Source: Watkins et al. electronic data http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7 (paper Sec. 15); Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014/2, 63-98, doi:10.5802/pmb.9 (https://pmb.centre-mersenne.org/item/10.5802/pmb.9.pdf).

last edited by David Renshaw at 2026-10-20 21:26:01 UTC · history